Curl
by Corinne Manogue
Copyright 2004 Corinne Manogue

In this worksheet you will look at some vector fields and calculate their curls to get a feel for what the curl is geometrically.

Define a three dimensional vector field for which the z-component is zero.  Assume that the cross-sections for constant values of z are identical and plot a cross-section as if the vectors are 2-dimensional.  (These special vector fields are chosen so that we can easily visualize what is going on.)  By looking at the vector field, decide at various points on the graph whether the curl is positive, negative, or zero.   Then us Mathematica to calculate the curl of the vector to check your geometric intuition.  The last vector field in the set is truly three-dimensional.

Clear[x,y,z]

Needs["VectorAnalysis`"]

Element[x,Reals]
Element[y,Reals]
Element[z,Reals]

v1={0,1,0}

Plot the vector field "2-d"--the third dimension is trivial

VectorPlot3D[v1,{x,-1,1},{y,-1,1},{z,-1,1},AxesLabel->{x,y,z},VectorPoints->5,VectorScale->Small,ViewPoint->{0,0,Infinity}]

You can calculate the curl of this function in your head...do you agree with Maple?

f1=Curl[v1,Cartesian[x,y,z]]

Now try another vector field:

v2={x,y,0}

VectorPlot3D[v2,{x,-1,1},{y,-1,1},{z,-1,1},AxesLabel->{x,y,z},VectorPoints->5,VectorScale->Small,ViewPoint->{0,0,Infinity}]

f2=Curl[v2,Cartesian[x,y,z]]

Now try another vector field:

v3={0,Exp[-y^2],0}

VectorPlot3D[v3,{x,-1,1},{y,-1,1},{z,-1,1},AxesLabel->{x,y,z},VectorPoints->5,VectorScale->Small,ViewPoint->{0,0,Infinity}]

f3=Curl[v3,Cartesian[x,y,z]]

Now try another vector field:

v4={0,Exp[-x^2],0}

Imagine putting a paddle wheel in the plane of the plot, it will turn if the curl is non-zero.  If you put a paddle wheel in the center, what will happen?  Where is the curl positive?  Where is it negative? (Recall, in a right handed coordinate system, you can use the right hand rule to determine which direction is positive.  Is positive into or out of the page?) Where is the curl zero?  Which vector component of the curl are we examining?

VectorPlot3D[v4,{x,-1,1},{y,-1,1},{z,-1,1},AxesLabel->{x,y,z},VectorPoints->5,VectorScale->Small,ViewPoint->{0,0,Infinity}]

f4=Curl[v4,Cartesian[x,y,z]]

Now try another vector field:

v5={-y,x,0}

VectorPlot3D[v5,{x,-1,1},{y,-1,1},{z,-1,1},AxesLabel->{x,y,z},VectorPoints->5,VectorScale->Small,ViewPoint->{0,0,Infinity}]

f5=Curl[v5,Cartesian[x,y,z]]

Now try another vector field:

v6={-y/(x^2+y^2),x/(x^2+y^2),0}
VectorPlot3D[v6,{x,-1,1},{y,-1,1},{z,-1,1},AxesLabel->{"x","y","z"},VectorPoints->5,VectorScale->Small,ViewPoint->{0,0,Infinity}]

f6=Simplify[Curl[v6,Cartesian[x,y,z]]]

Now define a "2-d" vector field of your own, plot it, and find its curl.

Define and plot some vectors in 3 dimensions.  Calculate the curl of these vector fields:

v7={x,y,z}

VectorPlot3D[v7,{x,-5,5},{y,-5,5},{z,-5,5},AxesLabel->{x,y,z},VectorPoints->5,VectorScale->Small]

By changing the orientation of the plot, you can view the plot in the plane of the coordinate axis (i.e. x-y plane, x-z plane, or y-z plane).  Convince yourself that each component of the curl is zero.  Now, calculate the curl.

f7=Curl[v7,Cartesian[x,y,z]]

Now choose your own 3-d vecor field, plot it, and find its curl: